Lecture4 - Electromagnetism Physics 15b Lecture#4 Divergence and Laplacian Purcell 2.72.12 What We Did Last Time Used Gausss Law on infinite sheet of

Lecture4 - Electromagnetism Physics 15b Lecture#4...

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1 Electromagnetism Physics 15b Lecture #4 Divergence and Laplacian Purcell 2.7–2.12 What We Did Last Time Used Gauss’s Law on infinite sheet of charge Uniform electric field E = 2 πσ above and below the sheet Electric field has energy with volume density given by Defined electric potential by line integral Electric field is negative gradient of electric potential Potential due to charge distribution: or Total energy of a charge distribution: u = E 2 8 π φ 21 = E d s P 1 P 2 = φ ( P 2 ) φ ( P 1 ) unit: erg/esu = statvolt E = −∇ φ φ = q j r j j = 1 N φ = dq r U = 1 2 ρφ dv
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2 Today’s Goals Introduce divergence of vector field How much “flow” is coming out per unit volume Translate Gauss’s Law into a differential (local) form Gauss’s Divergence Theorem connects the two forms Look in the energy again Equivalence of and Define the Laplacian = divergence of gradient Re-express Gauss’s Law with a Laplacian Study mathematical properties of Laplace’s equation Conclude with a Uniqueness Theorem U = 1 2 ρφ dv U = E 2 8 π dV Shrinking Gauss’s Law Charge is distributed with a volume density ρ ( r ) Draw a surface S enclosing a volume V Guass’s Law: Now, make V so small that ρ is constant inside V As we make V smaller, the total flux out of S scales with V Therefore: LHS is “ how much E is flowing out per unit volume Let’s call it the divergence of E E d a S = 4 π ρ dv V Total charge in V E d a S = 4 πρ V for very small V lim V 0 E d a S V = 4 πρ
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3 Divergence In the small- V limit, the integral depend on volume, but not on the shape We can use a rectangular box Consider the left ( S 1 ) and right ( S 2 ) walls Add up all walls: div E lim V 0 E d a S V = 4 πρ dx dy dz E ( x + dx , y , z ) E ( x , y , z ) S 1 S 2 E d a S 1 = E ( x , y , z ) ( ˆ x ) dydz E d a S 2 = E ( x + dx , y , z ) ˆ x dydz Sum = E x ( x + dx ) E x ( x ) ( ) dydz = E x x dxdydz E d a S = E x x + E y y + E z z V = ∇ ⋅ E ( ) V div E Gauss’s Law, Local Version We now have Gauss’s Law for a very small volume/surface Connects local properties of E with the local charge density Divergence has the easy form in Cartesian coordinates
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